So I completed 'The Man Who Cried Wolf' quest and leveled up. I then went into Markarth, since I read somewhere that after you level up the courier that is supposed to deliver Falk's letter will contatct you there. However, he never showed up. Is there a certain number of hours you have to wait? Any help would be appreciated.

Your level up needs to take you past 10 and your location doesn't matter. I've had the courier without leaving Solitude. The event that triggers the courier is the levelling up, but you must be at least level 10 after it happens. Since you can do the pre-requisite quest at 8 or less, you may need to level up some more.

I think I was level twenty-six when I did the quest, then leveled up two twenty-seven. Maybe I just need to wait a bit.

I finally got the letter. Did some digging around and it turns out I had to complete "Elisif's Tribute" first.

I don't do the Elisif's Tribute quest until after finishing Potema, so I'd never noticed they block each other. The shrine's a long way from Solitude, so it made more sense to finish the connected pair of quests before the trip.